Punita Batra


Research Summary:

During the last one year my research interest was focussed mainly on the following problems. I finished the problem of classification of irreducible finite dimensional modules for twisted multi loop Lie algebra. Let $\mathfrak g$ be a simple finite dimensional Lie algebra over ${\Bbb C}$. Let $L(\mathfrak g) = {\mathfrak g} \otimes {\Bbb C}[t_1^{{\pm 1} }, t_2^{{\pm 1} }, ...... t_n^{{\pm 1} }]$ be the multi loop algebra. Let $\mu$ be a Dynkin diagram automorphism of $\mathfrak g$ of order $k$. Let $\epsilon$ be a primitive $k$-th root of unity. We extend $\mu$ as an automorphism $\mu_1$ of $L(\mathfrak g)$ defined by

\begin{displaymath}\mu_1(Xt_1^{m_1}t_2^{m_2}...t_n^{m_n}) = {\epsilon}^{-m_1} \mu(X) t_1^{m_1}t_2^{m_2}...t_n^{m_n}.\end{displaymath}

Then fixed point subalgebra $L(\mathfrak g, \mu_1)$ of $L(\mathfrak g)$ under $\mu_1$ is a twisted multi loop algebra. The main aim of this paper was to construct a surjective Lie algebra homomprphism from $L(\mathfrak g, \mu_1)$ to a finite dimensional semisimple Lie algebra, so that the modules of $L(\mathfrak g, \mu_1)$ can be classified in terms of the finite dimensional semisimple Lie algebra. This work has beem completed and submitted for publication.

The second problem which i tried during my visit to TIFR in January 2003,was the complete reducibility of Integrable modules for the twisted affine Lie algebras. Let $\mathfrak g$ be a simple finite dimensional Lie algebra over $\Bbb C$. Let $\overline {\mathfrak g}^$ be the corresponding twisted affine Lie algebra and let $c$ be the canonical central element. In this problem, i have proved that any integrable module for $\overline {\mathfrak g}^$ with finite dimensional weight spaces where $c$ acts by non-zero scalars is completely reducible. Currently, I am trying to generalize the same problem for twisted affine super algebras.


Publications:

(With S. E. Rao) A new class of Representations of EALA Co-ordinated by Quantum Tori in two Variables, Canad. Math. Bull. Vol.45, 672-685(2002).

Preprints:

Representations of twisted multi loop Lie algebras.

Visits to other Institutes:

Visited TIFR, Bombay from June 26- July 3, 2002 and again from Dec26, 2002 - Jan 31, 2003.

Other Activities:

1) Gave five lectures on "Basics of Lie Algebras" at HRI in VSRP programme in May 2002.

2) Gave one lecture on Functions in Hindi in the Rajbhasha Programme organised by Dr V. Singh in May 2002.

3) Advised one research student on the project "Introduction to Lie algebras and Representation Theory" till July 2002.

4) Gave a first year graduate course "Introduction to Lie algebras and Representation Theory" in second semester 2003 at HRI.

5) Gave a seminar in March 2003 on the topic "Representations of Virasoro Algebra" at HRI.



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